34.07/10.24	YES
34.07/10.27	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
34.07/10.27	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
34.07/10.27	
34.07/10.27	
34.07/10.27	Termination w.r.t. Q of the given QTRS could be proven:
34.07/10.27	
34.07/10.27	(0) QTRS
34.07/10.27	(1) DependencyPairsProof [EQUIVALENT, 11 ms]
34.07/10.27	(2) QDP
34.07/10.27	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
34.07/10.27	(4) QDP
34.07/10.27	(5) QDPOrderProof [EQUIVALENT, 205 ms]
34.07/10.27	(6) QDP
34.07/10.27	(7) QDPOrderProof [EQUIVALENT, 70 ms]
34.07/10.27	(8) QDP
34.07/10.27	(9) PisEmptyProof [EQUIVALENT, 0 ms]
34.07/10.27	(10) YES
34.07/10.27	
34.07/10.27	
34.07/10.27	----------------------------------------
34.07/10.27	
34.07/10.27	(0)
34.07/10.27	Obligation:
34.07/10.27	Q restricted rewrite system:
34.07/10.27	The TRS R consists of the following rules:
34.07/10.27	
34.07/10.27	   a(x1) -> x1
34.07/10.27	   a(b(b(x1))) -> b(b(a(b(c(a(x1))))))
34.07/10.27	   b(c(x1)) -> x1
34.07/10.27	
34.07/10.27	Q is empty.
34.07/10.27	
34.07/10.27	----------------------------------------
34.07/10.27	
34.07/10.27	(1) DependencyPairsProof (EQUIVALENT)
34.07/10.27	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
34.07/10.27	----------------------------------------
34.07/10.27	
34.07/10.27	(2)
34.07/10.27	Obligation:
34.07/10.27	Q DP problem:
34.07/10.27	The TRS P consists of the following rules:
34.07/10.27	
34.07/10.27	   A(b(b(x1))) -> B(b(a(b(c(a(x1))))))
34.07/10.27	   A(b(b(x1))) -> B(a(b(c(a(x1)))))
34.07/10.27	   A(b(b(x1))) -> A(b(c(a(x1))))
34.07/10.27	   A(b(b(x1))) -> B(c(a(x1)))
34.07/10.27	   A(b(b(x1))) -> A(x1)
34.07/10.27	
34.07/10.27	The TRS R consists of the following rules:
34.07/10.27	
34.07/10.27	   a(x1) -> x1
34.07/10.27	   a(b(b(x1))) -> b(b(a(b(c(a(x1))))))
34.07/10.27	   b(c(x1)) -> x1
34.07/10.27	
34.07/10.27	Q is empty.
34.07/10.27	We have to consider all minimal (P,Q,R)-chains.
34.07/10.27	----------------------------------------
34.07/10.27	
34.07/10.27	(3) DependencyGraphProof (EQUIVALENT)
34.07/10.27	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
34.07/10.27	----------------------------------------
34.07/10.27	
34.07/10.27	(4)
34.07/10.27	Obligation:
34.07/10.27	Q DP problem:
34.07/10.27	The TRS P consists of the following rules:
34.07/10.27	
34.07/10.27	   A(b(b(x1))) -> A(x1)
34.07/10.27	   A(b(b(x1))) -> A(b(c(a(x1))))
34.07/10.27	
34.07/10.27	The TRS R consists of the following rules:
34.07/10.27	
34.07/10.27	   a(x1) -> x1
34.07/10.27	   a(b(b(x1))) -> b(b(a(b(c(a(x1))))))
34.07/10.27	   b(c(x1)) -> x1
34.07/10.27	
34.07/10.27	Q is empty.
34.07/10.27	We have to consider all minimal (P,Q,R)-chains.
34.07/10.27	----------------------------------------
34.07/10.27	
34.07/10.27	(5) QDPOrderProof (EQUIVALENT)
34.07/10.27	We use the reduction pair processor [LPAR04,JAR06].
34.07/10.27	
34.07/10.27	
34.07/10.27	The following pairs can be oriented strictly and are deleted.
34.07/10.27	
34.07/10.27	   A(b(b(x1))) -> A(x1)
34.07/10.27	The remaining pairs can at least be oriented weakly.
34.07/10.27	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
34.07/10.27	
34.07/10.27	   <<<
34.07/10.27	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 1A]] 	* 	x_1
34.07/10.27	>>>
34.07/10.27	
34.07/10.27	   <<<
34.07/10.27	 POL(b(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[0A, -I, 0A], [-I, 0A, 0A], [0A, 1A, 0A]] 	* 	x_1
34.07/10.27	>>>
34.07/10.27	
34.07/10.27	   <<<
34.07/10.27	 POL(c(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [-I, 0A, -I], [-I, 0A, -I]] 	* 	x_1
34.07/10.27	>>>
34.07/10.27	
34.07/10.27	   <<<
34.07/10.27	 POL(a(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [-I, 0A, -I], [0A, 1A, 0A]] 	* 	x_1
34.07/10.27	>>>
34.07/10.27	
34.07/10.27	
34.07/10.27	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
34.07/10.27	
34.07/10.27	   a(x1) -> x1
34.07/10.27	   a(b(b(x1))) -> b(b(a(b(c(a(x1))))))
34.07/10.27	   b(c(x1)) -> x1
34.07/10.27	
34.07/10.27	
34.07/10.27	----------------------------------------
34.07/10.27	
34.07/10.27	(6)
34.07/10.27	Obligation:
34.07/10.27	Q DP problem:
34.07/10.27	The TRS P consists of the following rules:
34.07/10.27	
34.07/10.27	   A(b(b(x1))) -> A(b(c(a(x1))))
34.07/10.27	
34.07/10.27	The TRS R consists of the following rules:
34.07/10.27	
34.07/10.27	   a(x1) -> x1
34.07/10.27	   a(b(b(x1))) -> b(b(a(b(c(a(x1))))))
34.07/10.27	   b(c(x1)) -> x1
34.07/10.27	
34.07/10.27	Q is empty.
34.07/10.27	We have to consider all minimal (P,Q,R)-chains.
34.07/10.27	----------------------------------------
34.07/10.27	
34.07/10.27	(7) QDPOrderProof (EQUIVALENT)
34.07/10.27	We use the reduction pair processor [LPAR04,JAR06].
34.07/10.27	
34.07/10.27	
34.07/10.27	The following pairs can be oriented strictly and are deleted.
34.07/10.27	
34.07/10.27	   A(b(b(x1))) -> A(b(c(a(x1))))
34.07/10.27	The remaining pairs can at least be oriented weakly.
34.07/10.27	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
34.07/10.27	
34.07/10.27	   <<<
34.07/10.27	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, -I, -I]] 	* 	x_1
34.07/10.27	>>>
34.07/10.27	
34.07/10.27	   <<<
34.07/10.27	 POL(b(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, -I, 1A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
34.07/10.27	>>>
34.07/10.27	
34.07/10.27	   <<<
34.07/10.27	 POL(c(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [0A, -I, 0A], [-I, -I, -I]] 	* 	x_1
34.07/10.27	>>>
34.07/10.27	
34.07/10.27	   <<<
34.07/10.27	 POL(a(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, -I, -I], [0A, 0A, -I], [0A, 0A, 0A]] 	* 	x_1
34.07/10.27	>>>
34.07/10.27	
34.07/10.27	
34.07/10.27	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
34.07/10.27	
34.07/10.27	   a(x1) -> x1
34.07/10.27	   a(b(b(x1))) -> b(b(a(b(c(a(x1))))))
34.07/10.27	   b(c(x1)) -> x1
34.07/10.27	
34.07/10.27	
34.07/10.27	----------------------------------------
34.07/10.27	
34.07/10.27	(8)
34.07/10.27	Obligation:
34.07/10.27	Q DP problem:
34.07/10.27	P is empty.
34.07/10.27	The TRS R consists of the following rules:
34.07/10.27	
34.07/10.27	   a(x1) -> x1
34.07/10.27	   a(b(b(x1))) -> b(b(a(b(c(a(x1))))))
34.07/10.27	   b(c(x1)) -> x1
34.07/10.27	
34.07/10.27	Q is empty.
34.07/10.27	We have to consider all minimal (P,Q,R)-chains.
34.07/10.27	----------------------------------------
34.07/10.27	
34.07/10.27	(9) PisEmptyProof (EQUIVALENT)
34.07/10.27	The TRS P is empty. Hence, there is no (P,Q,R) chain.
34.07/10.27	----------------------------------------
34.07/10.27	
34.07/10.27	(10)
34.07/10.27	YES
34.52/11.42	EOF